Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface. Seifert surfaces are also interesting in their own right, and the subject of considerable research.
Specifically, let L be a tame oriented knot or link in Euclidean 3-space (or in the 3-sphere). A Seifert surface is a compact, connected, oriented surface S embedded in 3-space whose boundary is L such that the orientation on L is just the induced orientation from S, and every connected component of S has non-empty boundary.
Note that any compact, connected, oriented surface with nonempty boundary in Euclidean 3-space is the Seifert surface associated to its boundary link. A single knot or link can have many different inequivalent Seifert surfaces. A Seifert surface must be oriented. It is possible to associate unoriented (and not necessarily orientable) surfaces to knots as well.
Other articles related to "seifert surface, seifert surfaces, seifert, surface":
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... a surface, of genus such that the boundary is either empty or is connected ... The Arf invariant of the framed surface is now defined Note that, so we had to stabilise, taking to be at least 4, in order to get an element of ... The Arf invariant of a framed surface detects whether there is a 3-manifold whose boundary is the given surface which extends the given framing ...
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“It was a pretty game, played on the smooth surface of the pond, a man against a loon.”
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